Large sets of Kirkman triple systems: the twelve remaining orders below 400

A large set of Kirkman triple systems, LKTS(v), is a partition of all $\binom{v}{3}$ triples of a $v$-point set into $v-2$ pairwise disjoint Kirkman triple systems (resolvable Steiner triple systems); only orders $v \equiv 3 \pmod{6}$ are admissible. The 2016 paper of Zheng, Chang and Zhou contributed direct difference-family constructions for seven new orders. The known direct and recursive constructions leave exactly twelve admissible orders below 400 open: 57, 87, 237, 249, 255, 267, 321, 327, 357, 375, 393, 399, on which every published direct and recursive method fails: none is divisible by 9, ten have composite $v-2$, for 375 and 399 the parameter $v-2 \in \{373, 397\}$ is prime congruent to $1 \pmod{4}$, and none satisfies $v \equiv 2 \pmod{7}$. The problem is to construct an LKTS(v) for each remaining order, or prove nonexistence; ten of the twelve are non-prime-power orders, on which the finite-field difference-family framework does not apply.

Background

A Steiner triple system $\mathrm{STS}(v)$ on a $v$-element point set $X$ is a collection of 3-element subsets (blocks, or triples) such that every 2-element subset of $X$ lies in exactly one block; it exists exactly for $v \equiv 1$ or $3 \pmod{6}$. The system is resolvable, and then called a Kirkman triple system $\mathrm{KTS}(v)$, if its blocks split into parallel classes — groups of $v/3$ pairwise disjoint triples that together cover all $v$ points; resolvability forces $v \equiv 3 \pmod{6}$. Kirkman's 1850 schoolgirl problem asked for a $\mathrm{KTS}(15)$, and Ray-Chaudhuri and Wilson proved in 1971 that a $\mathrm{KTS}(v)$ exists for every $v \equiv 3 \pmod{6}$.

A large set of disjoint Steiner triple systems partitions all $\binom{v}{3}$ triples of $X$ into $v-2$ pairwise disjoint $\mathrm{STS}(v)$; Lu (1983), completed by Teirlinck (1991), showed such large sets exist for every admissible $v \neq 7$. The Kirkman analogue is far harder. A large set of Kirkman triple systems, $\mathrm{LKTS}(v)$, is a partition of all $\binom{v}{3}$ triples into $v-2$ pairwise disjoint $\mathrm{KTS}(v)$. Beyond the trivial $\mathrm{LKTS}(9)$, the first example was Denniston's 1974 solution of Sylvester's problem, $\mathrm{LKTS}(15)$, followed by scattered small direct examples (Denniston 1979 and later authors) and a stock of recursive constructions — tripling and product-type recursions and more recent recursions through overlarge sets (analogues of large sets in which the member Kirkman systems, each omitting one point, partition all triples of a point set one larger) and related designs — whose output orders are typically divisible by 9 or tied to prime-power parameters. Existence of an $\mathrm{LKTS}(v)$ for every admissible $v$ is one of the best-known open problems of combinatorial design theory, and the known spectrum is sparse: below order 200 only the orders 57 and 87 remain open, and ten further orders between 200 and 400 remain open, twelve in all below 400.

The source paper (Zheng, Chang and Zhou, 2016) constructs $\mathrm{LKTS}(q+2)$ directly when $q$ is a prime power with $q \equiv 7$ or $19 \pmod{24}$, using difference families over the cyclic group $\mathbb{Z}_{q-1}$: base blocks whose group translates generate the triples, organized into starter parallel classes — base block sets whose developed orbits form parallel classes — with several disjoint starters needed to reach all $q = v-2$ systems. The method proved $\mathrm{LKTS}(v)$ for $v \in \{69, 141, 165, 213, 285, 309, 333\}$; the remaining admissible orders below 400 were left unsettled, chiefly orders whose parameter $q = v-2$ is a prime power $\equiv 1 \pmod{4}$ — where the search for two or more starter parallel classes over $\mathbb{Z}_{q-1}$ is computationally difficult and no new orders were obtained — and non-prime-power orders ($v-2$ composite), on which the entire finite-field difference-family framework does not apply.

Problem Statement

The open cases are: for each of the twelve admissible orders

$$v \in \{57, 87, 237, 249, 255, 267, 321, 327, 357, 375, 393, 399\},$$

decide whether a large set of Kirkman triple systems LKTS($v$) exists — that is, construct $v-2$ pairwise disjoint resolvable Steiner triple systems on a common $v$-point set that jointly partition all $\binom{v}{3}$ triples — or prove that none exists. These twelve are exactly the admissible orders below 400 not covered by any of the direct or recursive constructions cited below. Ten of the twelve orders (all but 375 and 399) are non-prime-power orders, with $v-2$ composite, on which the finite-field difference-family framework of the source paper does not apply; for the remaining two, the parameter $q = v-2 \in \{373, 397\}$ is prime with $q \equiv 1 \pmod{4}$, the regime in which no efficient search for two or more starter parallel classes over $\mathbb{Z}_{q-1}$ is currently available.

References

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